Internal problem ID [8588]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 252.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {\left (x^{2} y-1\right ) y^{\prime }-x y^{2}=-1} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 963
dsolve((x^2*y(x)-1)*diff(y(x),x)-(x*y(x)^2-1)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {4^{\frac {2}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {2}{3}}+\left (\left (-c_{1} +80\right ) x^{7}-160 x^{4}+80 x \right ) 4^{\frac {1}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {1}{3}}+\left (c_{1}^{2}-80 c_{1} \right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}}{x^{2} 4^{\frac {2}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {2}{3}}+\left (c_{1} x^{4}-4^{\frac {1}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {1}{3}}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\ y \left (x \right ) &= \frac {4^{\frac {2}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (2 i 4^{\frac {1}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {1}{3}}+\left (i-\sqrt {3}\right ) x c_{1} \right ) x \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )}{x^{2} 4^{\frac {2}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (2 i 4^{\frac {1}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {1}{3}}+\left (i-\sqrt {3}\right ) x^{4} c_{1} \right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\ y \left (x \right ) &= \frac {\left (i-\sqrt {3}\right ) 4^{\frac {2}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {2}{3}}+x \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) \left (2 i 4^{\frac {1}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {1}{3}}+x c_{1} \left (\sqrt {3}+i\right )\right )}{\left (i-\sqrt {3}\right ) x^{2} 4^{\frac {2}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {2}{3}}+\left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) \left (2 i 4^{\frac {1}{3}} {\left (\left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right ) c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2}\right )}^{\frac {1}{3}}+x^{4} c_{1} \left (\sqrt {3}+i\right )\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 36.312 (sec). Leaf size: 506
DSolve[(x^2*y[x]-1)*y'[x]-(x*y[x]^2-1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}{-1+6 c_1}-\frac {x^2}{\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}+x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}{-2+12 c_1}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}+x \\ y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}{-2+12 c_1}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}+x \\ y(x)\to x \\ \end{align*}